Role of back-diffusion studied by computer simulation - Springer Link

Role of Back-Diffusion Studied by Computer Simulation ¨ M. HILLERT L. HOGLUND, and M. SCHALIN The solidification of Fe-Cr-C alloys was simulated with a program that allowed realistic treatment of diffusion in the liquid and the primary solid phase. It interacts with a thermodynamic databank and can evaluate the heat extraction continuously. Solidification under a constant rate of heat extraction can be simulated. In agreement with previous experience, diffusion of carbon in the fcc phase was found to be very effective, and it can be approximated by an infinitely fast diffusion. The same holds for carbon and chromium in the liquid. The back-diffusion of chromium in the fcc phase was examined in detail. Three different geometries were tested: the planar and cylindrical cases, as well as a new model based on two coupled cylinders. The simulation program allowed the effect of back-diffusion after the end of solidification to be included, and it was found to be important for the segregation ratio.

I. INTRODUCTION

SOLIDIFICATION of alloys generally occurs with segregation, and this phenomenon may have a strong effect on the progress of the solidification. There has, thus, been a considerable interest in modeling segregation. One must distinguish between two types. The present work only concerns microsegregation, whereas the effects of macrosegregation will be completely neglected. The simplest case of microsegregation is modeled by considering a small part of a dendrite and the surrounding liquid. Often one applies a planar model, which is an approximation but may be justified as a reasonable simplification of a planar arrangement of a set of densely spaced secondary arms. Another approximation may be to model the thickening of a single secondary arm by considering a cylindrical geometry. More ambitious treatments of the geometry of dendrites, including coarsening, have been attempted by Sundarraj and Voller,[1] for instance, but will not be discussed further in the present article. A crude but very powerful method of estimating microsegregation was proposed by Gulliver in 1913[2] and was again applied by Scheil in 1942.[3] The main approximations were to treat the rate of diffusion in the liquid as infinite but negligible in the solid phases. With that model, one does not have to specify the geometry. An additional approximation was to treat the liquidus and solidus in the phase diagram as straight lines. That approximation is no longer so important, due to the high speed of numerical computer calculations. Computer programs for the simulation of microsegregation in alloys are now common. They can be coupled to thermodynamic databases from which the solid/ liquid equilibrium can be calculated repeatedly during the simulation. They can, thus, be used for alloys with a nonlinear solidus and liquidus and with more than two components (for example, Reference 4). It was recently shown[5] that a simulation program can even be obtained directly from a thermodynamic databank with facilities for stepwise calculations of equilibria. Only minor modifications were required. Another important improvement was the introduction of ¨ M. HILLERT, Professor, L. HOGLUND, Research Associate, and M. SCHALIN, Graduate Student, are with the Department of Materials Science and Engineering, KTH,SE-10044 Stockholm, Sweden. Manuscript submitted September 15, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A

more realistic treatments of diffusion, which was done by modifying the analytical expression of Gulliver and Scheil. It was first done by Broady and Flemings,[6] and their treatment was later improved by Cline and Kurz[7] and Ohnaka.[8] The resulting equations are simple to use, but Kobayashi[9] finally gave an exact treatment of the model proposed by Broady and Flemings and found that all three equations proposed by the previous authors could sometimes give very erroneous results. Even though Kobayashi gave an exact treatment, it must be realized that the basic assumptions are not very good. For instance, a parabolic growth law was assumed. More recently, numerical methods have been introduced in order to improve the treatment of diffusion, and several computer programs have been constructed. Of course, such programs can also be coupled to thermodynamic databanks. Usually, such programs take diffusion in the solid into account (the so-called back-diffusion), but they still approximate the diffusion in the liquid as infinitely fast. In the present article, a more general type of program for the simulation of diffusional phase transformations, DICTRA,[10] will be applied in an investigation of the effect of diffusion on the progress of solidification in alloys with two alloying elements, one very mobile and the other more sluggish. That program allows diffusion to be treated in an ambitious way in the liquid phase as well as in the solid material. It is different from a program recently used by Lacaze and Lesoult[11] in a similar study, in that it interacts with a thermodynamic databank for the calculation of the solid/liquid equilibrium and for the evaluation of the change in heat content at each step. It can, thus, simulate solidification under a constant rate of heat extraction, for instance. II. EXPERIMENTAL INFORMATION The segregation ratio (maximum content/minimum content) of Cr in Fe-Cr-C alloys with 1 pct Cr has been studied as a function of the C content by several authors.[12,13,14] It was found that it is about 1.5 at low C contents, it increases to about 5 at 1.6 pct C, and it then decreases again. The increase was explained by Fredriksson and Hellner[14] as an effect of a strong interaction between Cr and C. That interaction is stronger in the liquid phase than in the solid (fcc phase, except for very low C contents). The decrease was explained by the appearance of a eutectic at the end of VOLUME 30A, JUNE 1999—1635

the solidification process. The eutectic reaction starts sooner, i.e., at a lower fraction of solid, the higher the C content is, and it reverts the direction of segregation. Fredriksson and Hellner made a quantitative prediction using analytical methods and found that it was important to take into account back-diffusion of Cr as well as C in the solidified material. However, their calculations involved a number of approximations that can now be avoided using numerical simulation. A series of calculations were then performed in order to study in more detail the role of back-diffusion and noninfinite diffusion in the liquid. The presence of the bcc phase at low C contents was neglected, for reasons to be explained later. In the experiments by Fredriksson and Hellner,[14] the ingots were 150-mm square, 300-mm long, and weighed about 50 kg. The total solidification time of an ingot was 4 or 8 minutes, depending on the cooling conditions. In their calculations, they used the same values for the local solidification time and, lacking better information, their values were accepted in the present work. The secondary dendrite arm spacing varied between 0.1 and 0.2 mm, depending on the C content, the position in the ingot, and the solidification time. It should, further, be realized that the spacing has probably varied considerably during solidification due to coarsening. A constant value of 0.15 mm was adopted in the present work, in view of all the uncertainties. The thermodynamic description of the ternary system Fe-Cr-C was taken from Hillert and Qiu[15] and the diffusion data from ¨ Jonsson.[16,17] A series of alloys was considered, with 1 pct Cr and a C content varying from 0 to 3 pct. Throughout this article, “percent” implies mass percent. The normalized distance in diagrams measures the amount of material between any position and the origin of solidification. The amount of material is expressed through the content of Fe 1 Cr, expressed in mol, neglecting the C content.

III. SIMULATIONS A. Gulliver–Scheil Approximations The first set of calculations was made with the classical Gulliver–Scheil approximation, i.e., no diffusion in the solid and infinite diffusion in the liquid. For this purpose, the ThermoCalc databank[18] was used according to the instructions for Gulliver–Scheil segregation given by Sundman and Ansara.[19] The solidification results in a primary precipitation of the Fe-rich fcc phase, called austenite, and a eutectic reaction to fcc 1 cementite (Fe,Cr)3C. The result is presented in Figure 1, and it is shown that, by practical reasons, these calculations were stopped when about 1 pct liquid remained. The region with the higher Cr content represents the eutectic. The diagram demonstrates the self-evident fact that the higher the carbon content is, the earlier the eutectic starts. There are two interesting features. The Cr content of the eutectic structure decreases during the process. Thus, there is a negative segregation of Cr during the eutectic reaction, and that reaction evidently follows the eutectic in the direction toward the Fe-C binary side of the phase diagram. Furthermore, the alloy reaches the eutectic at a lower Cr content, the higher the C content of the alloy is. It should 1636—VOLUME 30A, JUNE 1999

Fig. 1—Segregation curves for Fe-Cr-C alloys with 1 mass pct Cr, calculated with the Scheil approximation. Regions with high Cr contents are eutectic. The curves show positive segregation during the primary precipitation of fcc and negative segregation during eutectic solidification.

be remembered that the average Cr content was 1 pct in all the alloys. The result of this kind of simulation is independent of the cooling rate. It is, thus, possible to evaluate the cooling rate afterward in order to fit the experimental solidification time. However, if one wants to get the same solidification time, one needs different cooling rates for different alloys, because the solidification range, measured in kelvin, varies with the C content. In order to compare different alloys, it may be better instead to assume that the rate of heat extraction is constant during the whole process and is the same for all alloys, independent of the variations in heat capacity. In order to use that assumption, it is required that one evaluate the change of enthalpy during the solidification process. For such purposes, a simulation program was constructed by Schalin,[20] based on the ThermoCalc databank, by adding an evaluation of the enthalpy of all the zones solidified at various steps in the simulation. The enthalpy change of the alloy with 1.5 pct C was evaluated during the whole process of solidification. That value was combined with a solidification time of 8 minutes, reported for some of the experiments. A rate of heat extraction of 40 J/mols was, thus, obtained, and it was then used for all the alloys in the later calculations with DICTRA, where the rate of cooling affects the result due to finite diffusion. The DICTRA program can also evaluate the change in enthalpy and can be used for simulations under a constant rate of heat extraction. That feature was utilized in the present work. B. Back-Diffusion of Carbon The most severe approximation in the previous set of calculations is probably the neglect of back-diffusion of C into the solid material. The second set of calculations was made with very fast diffusion of C in the whole system and of Cr in the liquid, using the DICTRA program. Those METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 2—Segregation curves calculated for very fast diffusion of C in the whole system and of Cr in the liquid. The simulation was interrupted when the eutectic reaction started. The regions of high-Cr contents represent the liquid volume at that moment.

results are presented in Figure 2. It is evident that the backdiffusion of C into the previously solidified material decreases the amount of C remaining in the liquid to such a large extent that the eutectic reaction never occurs in the alloys of less than 2.0 pct C. It should be emphasized that the DICTRA simulations were stopped at the start of the eutectic reaction because that program cannot handle the diffusion through a layer of two eutectic phases growing side by side. Primarily, each curve shows the Cr profile of the whole system at the start of the eutectic reaction. The curve is horizontal in the liquid phase, due to the assumption of infinitely fast diffusion there. The Cr content of the first eutectic structure to form was calculated afterward from the liquid composition using the phase diagram and is marked with a point on the top of a dashed line. The eutectic reaction will, then, result in a negative segregation, as already illustrated in Figure 1, and the average of such a curve would be represented, approximately, by the horizontal part of the curves in Figure 2. It may be noted that the Cr content in the diagrams is expressed in a composition variable defined as uCr 5 NCr / (NFe 1 NCr), because that variable is independent of changes in the C content. Expressed in this way, the minimum Cr content, which is also the content of the first solid to form, is not affected by the C diffusion, the reason being that Cr diffusion in the solid was neglected in both calculations.

C. Back-Diffusion of Carbon and Chromium A third set of simulations was made, with a realistic treatment of back-diffusion of C and Cr in the solid Fe phase, but still with infinite diffusion in the liquid. For such a calculation, one must know the geometry, and, initially, a planar system with a length of 0.15/2 mm was considered. METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 3—Variation of C activity at the end of the primary precipitation, calculated for realistic rates of diffusion of C and Cr in the solid and infinite diffusion in the liquid. The simulation was interrupted when the eutectic reaction started.

Fig. 4—Segregation curves for Cr in the same simulation as Fig. 3. The squares represent the Cr content of the first eutectic to form.

Those results are presented in Figures 3 and 4, and the curves represent the distribution of C and Cr at the moment when the eutectic reaction starts. The activity is plotted for C, and it is evident that it is very uniform in the whole system. It may be concluded that the realistic rate of C diffusion could just as well be replaced by an infinitely rapid diffusion, if that would simplify the calculations. Comparison of Figure 4 to Figure 2 demonstrates that the back-diffusion of Cr has only a slight effect on the start of the eutectic reaction. The most evident difference is that the curves for 2 pct C show VOLUME 30A, JUNE 1999—1637

Fig. 5—Segregation curves calculated for realistic rates of diffusion in both phases. The squares are defined as in Fig. 4.

Fig. 6—Segregation curves calculated for a two-cylinder model with realistic rates of diffusion in both phases.

that the eutectic reaction starts a little sooner and results in a eutectic of much less Cr, when back-diffusion of Cr is considered. D. Finite Diffusion in Liquid A fourth set of simulations was made, with a realistic treatment of diffusion in liquid as well as solid. Those results are presented in Figure 5, which is very similar to Figure 4. It is evident that even the realistic rate of diffusion in the liquid is sufficient to eliminate the composition differences between secondary dendrite arms. It is, thus, confirmed that it is a very good approximation to assume infinitely rapid diffusion of Cr, as well as C, in the liquid. E. Effect of Geometry Then, a set of calculations was made in an attempt to simulate the geometry more realistically. The calculation is supposed to simulate the thickening of a secondary arm of a dendrite. A cylindrical cell or, even better, a hexagonal cell may, thus, be used where the solid starts to form as a thin thread in the center of parallel arms. However, toward the end, the remaining liquid would then be described as a thin layer around the periphery with a very large interface to the solid. That may be realistic in solidification under a very strong temperature gradient (for example, refer to Ueshima et al.[21]). In other cases, it may be more realistic again to use a cylindrical geometry, but to place the remaining liquid as a rod in the center. This can be accomplished by using two cylindrical cells and letting the reaction start in the center of one and finish at the center of the other. This is similar to two models by Ueshima et al., developed from their hexagonal model. Material must be allowed to transfer between the peripheries of the two cells according to the rate of diffusion. The geometry of the system can be adjusted by choosing cylinders of different radii, but that 1638—VOLUME 30A, JUNE 1999

Fig. 7—Segregation curves calculated for a one-cylinder model with realistic rates of diffusion in both phases.

would give a nonphysical break in the composition profiles at the boundary between the cylinders. One may even choose a cylinder and a sphere. In the present case, two cylinders of the same size were chosen. The DICTRA program could be used directly for these calculations. The radii of the two cylinders were taken as 0.15/4 mm, and realistic diffusion coefficients were used for both phases. Those results are presented in Figure 6 and may be compared to the onecylinder model in Figure 7. The radius of that cylinder was taken as !2 3 0.15/4 mm, in order to make the volume equal to that of the two-cylinder model, with a radii equal to 0.15/4 mm. METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 8—The segregation ratio as a function of the C content of the alloy, using the planar model. The dotted curve was evaluated with the maximum Cr content taken from the fcc phase at the start of the eutectic reaction. The dashed curve was taken from the first eutectic to form (squares in Fig. 5).

Comparison between Figures 5 and 7 shows only small differences between the planar model and the one-cylinder model. Comparision between Figures 6 and 7 shows a striking difference between the two cylindrical models. The twocylinder model has not delayed the eutectic reaction as much, and that reaction, thus, occurs in alloys with a lower C content. This is related to the fact that the two models do not really differ until after half of the system (i.e., the first cylinder in the two-cylinder model) has solidified, and the difference grows more pronounced the further the solidification proceeds. On the other hand, it is interesting to note that the Cr content of the eutectic is almost the same when there is eutectic in both models. Evidently, it does not make much sense to use the two-cylinder model unless one is particularly interested in studying the effect of segregation toward the end of the solidification. IV. EVALUATION OF THE SEGREGATION RATIO The segregation ratio is defined as the ratio between the maximum and minimum contents of the alloying element, expressed in mass percent. The minimum value was always found in the position where the solidification had started. When evaluating the segregation ratio, the planar model was first used. Maximum values were evaluated from Figure 5 using two methods. One was based on the points evaluated for the first eutectic to form (squares in Figure 5); refer to the dashed line in Figure 8. The other was based on the Cr content in the liquid at the start of the eutectic reaction; refer to the full line in Figure 8. In order to get smooth curves, calculations were also made for many more C contents than those shown in the diagram. In Figure 8, the results are compared to experimental data compiled by Fredriksson and Hellner.[14] METALLURGICAL AND MATERIALS TRANSACTIONS A

The calculations resulted in a single curve at low C contents, where there is only the fcc phase. It shows increasing values as the C content of the alloy is increased, because C increases the solidification range and, thus, the segregation. That curve stops at the C content where a eutectic first appears. In such an alloy, the maximum Cr content is obtained in the eutectic structure (dashed curve), and that falls at a much higher value because of the high Cr content in the eutectic carbide being higher than that in the fcc phase. The curve based on the liquid composition (full line) gives values almost as high. These curves go to lower values at increasing C contents, because there the eutectic reaction starts earlier, i.e., before the segregation has developed as much. In addition, a curve was calculated for the maximum Cr content in the last fcc phase to form before the onset of the eutectic reaction. Naturally, that curve joins with the first curve (dotted curve in Figure 8). Considering the large scatter between various experimental studies, the low-C curve describes the data up to 1.5 pct reasonably well, but it may be complained that the curve falls above all the experimental points close to the left-hand side of the diagram. This was also true for a curve calculated by Fredriksson and Hellner, and they mentioned the possibility that this could be due to the solidification resulting in bcc instead of fcc phase, the bcc phase having a partition coefficient much closer to unity than that of the fcc phase. However, they argued that the experimental segregation is definitely larger than expected for the bcc phase and proposed that, by kinetic reasons, the solidification results in fcc phase although the phase diagram indicates that the solidification should start by precipitation of bcc phase at C contents up to about 0.5 pct. It may here be added that, if bcc phase were formed, it should transform to the fcc phase by a peritectic reaction later during the solidification process, if the C content is higher than 0.18 pct. Since it was not the intention to study the peritectic reaction, the present simulations were carried out with complete neglect of the possibility that the bcc phase may play a role at early stages of solidification at low C contents. Fredriksson and Hellner made the same assumption in their calculations. At very high C contents, the curve obtained from the evaluated composition of the first eutectic (dashed line) shows the best agreement with the experimental points. As the C content is lowered, the experimental points fall below that curve and approach the dotted curve, which was evaluated from the fcc phase with the highest Cr content. That can be explained by the fact that the amount of eutectic grows smaller and smaller. It may, thus, be more difficult to hit the eutectic areas with measurements. Furthermore, small regions of eutectic may even have disappeared by continued back-diffusion during the cooling of the ingots after completed solidification. The results of the two cylindrical models are compared to the planar model in Figure 9. In order to simplify the diagram, the curves obtained from the eutectic composition were here omitted (dashed line in Figure 8). At high-C contents, there is practically no difference, the reason being that the simulated segregation stops when there is still a considerable amount of liquid present, due to the start of the eutectic reaction. However, the curves extend to different low-C contents. It is evident that the eutectic reaction starts VOLUME 30A, JUNE 1999—1639

Fig. 9—Comparison between segregation ratios obtained with different geometries.

Fig. 10—Segregation ratios calculated after cooling to low temperatures. Only alloys without eutectic reaction are included.

earlier according to the two-cylinder model and later according to the one-cylinder model, the reason being the different effects of the geometries on back-diffusion toward the end of solidification. The most striking difference is found in the low-C range. These curves are based on the Cr content in the last fcc phase, and the fact that the twocylinder model yields the highest values again reflects the fact that back-diffusion toward the end of solidification is less in this model, because the area of the boundary between liquid and solid is less. There would, thus, be less backdiffusion. For the same reason, the one-cylinder model yields lower values than the planar model because that area is larger in this model. This result is in qualitative agreement with Lacaze and Lesoult,[11] who found that the spherical model gave more back-diffusion than the planar model. It is evident that the amount of back-diffusion toward the end of solidification is very important. It thus seems very likely that additional back-diffusion occurring after the end of solidification should also be very important. As a consequence, the simulation was extended to include the cooling to low temperatures. Those results will be presented in the following section. It is interesting to note that Lacaze and Lesoult[11] took the opposite view, stressing that backdiffusion after the end of solidification should not have a large influence on most of the distribution curve.

serious. Results for the three geometries are presented in Figure 10 and are there compared to the experimental results below 1.5 pct C. These simulations were not performed at the higher C contents, where the eutectic appears, because, as already mentioned, the DICTRA program cannot handle diffusion through a zone with a two-phase mixture. Compared to Figure 9, all three curves in Figure 10 have moved to much lower values due to the additional backdiffusion. The planar model now yields excellent agreement with the data reported by Fredriksson and Hellner,[14] even at the lowest C contents. It should be emphasized that the relative effect of additional back-diffusion can be expected to be larger when the solidification range is smaller. When discussing what geometric model one should choose for a realistic simulation, it is evident that the role of additional back-diffusion after the end of solidification must be taken into account. Judging from the present results, the planar model seems to reproduce the experimental data best. However, a definite recommendation should not be made without further tests. It is interesting that, on different grounds, Lacaze and Lesoult concluded that segregation is better simulated with a spherical model than with a planar one. One may, thus, reach different conclusions, depending on how the extent of segregation is defined or judged. VI. DISCUSSION

V. BACK-DIFFUSION AFTER THE END OF SOLIDIFICATION The simulations with the DICTRA program could be directly extended to lower temperatures as long as carbide does not form. Such simulations were, thus, performed but were interrupted when carbide appeared. The same rate of heat extraction was used as during solidification, although it should decrease as the temperature falls. However, most of the additional back-diffusion will occur soon after the end of the solidification, and this approximation may not be 1640—VOLUME 30A, JUNE 1999

Fredriksson and Hellner[14] studied the segregation ratio in Fe-Cr-C alloys by experimental measurements as well as by calculation. For high C contents, they assumed equilibrium distribution of C and used a Gulliver–Scheil calculation for Cr. Those results are most similar to the curve calculated in the present work using the composition of the eutectic liquid. For the low C content, Fredriksson and Hellner[14] made calculations based on the planar segregation model by Broady and Flemings,[6] which is an improvement of the METALLURGICAL AND MATERIALS TRANSACTIONS A

Gulliver–Scheil equation because it takes back-diffusion into account, although in a rough way. Their curve starts from the same point on the left-hand side as the curve obtained for the planar model in the present work. However, it rises more quickly with increasing C contents than the present curve and there it deviates more from the experimental data and from the present curve. It may be that the Broady–Flemings equation has not predicted the full effect of back-diffusion during solidification. The difference is even larger when the effect of back-diffusion during the cooling after the end of solidification is taken into account. The Broady–Flemings equation was not intended to take that effect into account. In summary, it should be stated that Fredriksson and Hellner,[14] with their more approximate calculations, were able to explain the main features of the variation of the segregation ratio with the C content, and their explanation of the maximum as an effect of the eutectic reaction is certainly correct. The present simulations have resulted in some improved agreement with experiments, if one chooses the planar model, and takes into account back-diffusion during cooling. It may be emphasized that such simulations can now be carried out without much more work than more approximate calculations. However, it is still an open question as to which geometrical model one should choose. In the process of the present work, the effect of diffusion of C and Cr in liquid and solid were explored. In line with previous experience, it was concluded that one can get good results by approximating the rates of diffusion of C and Cr in the liquid and C in the fcc phase as infinitely fast. For Cr, it is essential to treat back-diffusion in the fcc phase in an ambitious way and, also, back-diffusion during cooling after the end of solidification should not be neglected. The present simulations were carried out with a rather general type of program for diffusional transformations (DICTRA). It was applied to the condition of constant rate of heat extraction, which implies that the rate of cooling is considerably higher after than during solidification. Even

METALLURGICAL AND MATERIALS TRANSACTIONS A

so, the role of back-diffusion after the end of solidification was found to play an important role. ACKNOWLEDGMENTS Thanks are due to CAMPADA for financial support of this project and to KTH for a grant to one of the authors (LH). REFERENCES 1. S. Sundarraj and V.R. Voller: Int. J. Heat Mass Transfer, 1993, vol. 36, pp. 713-23. 2. G.H. Gulliver: J. Inst. Met. 1913, vol. 9, pp. 120-57. 3. E. Scheil: Z. Metallkd., 1942, vol. 34, pp. 70-72. 4. S.-W. Chen and Y.A. Chang: Metall. Trans. A, 1991, vol. 22A, pp. 267-71. ˚ 5. A. Jansson and B. Sundman: TRITA MAC 610, Royal Institute of Technology, Stockholm, 1997. 6. H.D. Brody and M.C. Flemings: Trans. AIME, 1966, vol. 236, pp. 615-24. 7. T.W. Clyne and W. Kurz: Metall. Trans. A, 1981, vol. 12A, pp. 965-71. 8. I. Ohnaka: Tetsu-to-Hagane´, 1984, vol. 70, pp. 913-15. 9. S. Kobayashi: J Cryst. Growth, 1998, vol. 88, pp. ˚ 87p96. ¨ ¨ 10. J.-O. Andersson, L. Hoglund, B. Jonsson, and J. Agren: in Fundamentals and Applications of Ternary Diffusion, G.R. Purdy, ed., Pergamon Press, New York, NY, 1990, pp. 153-63. 11. J. Lacaze and G. Lesoult: Iron Steel Inst. Jpn. Int., 1995, vol. 35, pp. 658-64. 12. P.M. Zhurenkov and I.N. Golikov: Met. Sci. Heat Treatment Met., 1964, pp. 293-97. 13. R.D. Doherty and D.A. Melford: J. Iron Steel Inst., 1966, vol. 204, pp. 1131-43. 14. H. Fredriksson and L. Hellner: Scand. J Metall., 1974, vol. 3, pp. 61-68. 15. M. Hillert and C. Qiu: Metall. Trans. A, 1991, vol. 22A, pp. 2187-98. ¨ 16. B. Jonsson: Z. Metallkd., 1994, vol. 85, pp. 502-09. ¨ 17. B. Jonsson: Trita-Mac 562, KTH, Stockholm, 1994. 18. B. Sundman, B. Jansson, and J.O. Andersson: CALPHAD, 1985, vol. 9, pp. 153-90. 19. B. Sundman and J. Ansara: in The SGTE Casebook, K. Hack, ed., The Institute of Materials, London, 1996, pp. 94-98. 20. M. Schalin: KTH, Stockholm, unpublished work, 1998. 21. Y. Ueshima, S. Mizoguchi, T. Matsumiya, and H. Kajioka: Metall. Trans. B, 1986, vol. 17B, pp. 845-59.

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